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Journal of Vibration and Control

DOI: 10.1177/107754639900500203 1999; 5; 195 Journal of Vibration and Control

I. Basdogan and T.J. Royston High-Precision Optical Positioning Systems

A Theoretical and Experimental Study of the Vibratory Dynamics of

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A Theoretical and Experimental Study of the VibratoryDynamics of High-Precision Optical Positioning Systems

I. BASDOGAN*

T. J. ROYSTON* *

Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, USA

(Received 24 November 1997; accepted 12 February 1998)

Abstract: At the Advanced Photon Source (APS), a state-of-the-art synchrotron radiation facility at ArgonneNational Laboratory (ANL), high-precision optical positioning systems are needed to conduct a wide rangeof experiments using the high-brilliance X-ray beam. Precision may be compromised by low-level, low-frequency vibrations from flow-structure interactions in the cooling systems and from facility-based distur-bances propagating through the floor. To predict the vibratory response of the positioning systems, a linearizedmultibody formulation has been developed. It has been applied to specific example cases—an optical tableand a mirror support system—used at the experimental stations of the APS. Comparisons of resonant fre-quency and mode shape predictions based on the theoretical formulation with experimental measurements il-lustrate the crucial importance of properly modeling the kinematic joints and components that comprise thesemultibody structures. Improved experimental and theoretical methods have been introduced to estimate theirdynamic properties. The results obtained by theory compare well with experimental findings. The proposedmethodology is precise and generic in predicting the coupled multidimensional, multi-degree-of-freedom vi-bratory motion of the positioning systems for the given positioning configurations. It is easily adaptable tonumerous systems at the APS and similar facilities.

Key Words: High-precision positioning, vibration control, optical systems

1. INTRODUCTION

High-precision positioning is required in numerous applications, including industrialmicrofabrication processes, optics, metrology, and physics research. Specific examplesinclude the fabrication of semi-conductor-based circuits with features on the submicron level

(Brochet and Bernard, 1991), scanning electron microscopes (Jennings and Willson, 1991),telescopes (Dyck, Berg, and Harris, 1986), and linear colliders (Baklakov et al., 1993). In

all of these cases, vibratory motion can limit the level of positioning precision and stability.Structural resonant responses may occur due to environmental loads such as wind, seismicmotion, or human activity. Furthermore, excitation sources can be in the form of mechanical

* currently at NASA Jet Propulsion Laboratory** corresponding author [email protected]

Journal of Vbration and Contml, 5: 195-216, 1999© 1999 Sage Publications, Inc.

195



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components such as motors, pumps, and rotating machinery. The prediction and control of theresponse to these disturbances is fundamental to the design and operation of high-precisionequipment.

At the Advanced Photon Source (APS), a state-of-the-art facility at Argonne NationalLaboratory (ANL), high-brilliance X-ray beams must be precisely directed throughexperimental stations via optical instruments to distant targets. Target distances up to 30meters with focused beam spot sizes as small as 1 mm in width are needed to support researchin a wide range of technical fields, including structure of materials, X-ray imaging, andbiomedical research. Stability of the beam may be compromised by low-level, low-frequencyvibrations caused by flow-structure interactions in the cooling systems of the optical devicesand by facility-bome disturbances propagating through the floor. The support systems usedto position synchrotron radiation instruments must meet multi-degree-of-freedom positioningcapability under sometimes heavy load conditions with accuracy on the order of microns andfractions of microradians.

Detailed vibratory studies are needed to understand and improve the performance ofthese positioning systems under the effect of various excitation sources. To predict theoverall system dynamics, it is necessary to have accurate theoretical models of the complexmultidimensional linkages and joints in the system. Having an accurate system model enablesthe designer to investigate the dynamic behavior of these structures and perform parametricdesign studies. Such a model could also provide information for a better understanding ofthe design principles to maintain functionality and ultimately to increase the capability of theequipment in a cost-effective way.

In this study, a linearized multibody dynamic formulation has been combined withexperimental methods to predict the dynamic behavior of high-precision optical positioningsystems. This approach has been applied to an optical table and a mirror support system usedat the experimental stations of the APS. Unlike earlier vibration studies on high-precisionstructures (Brochet and Bernard, 1991; Jennings and Willson, 1991; Dyck, Berg, and Harris,1986; Baklakov et al., 1993; Doyle et al., 1995; Doyle, 1993; Redding and Breckenridge,1991; Anees, 1995; Laughlin, 1995; Jendrzejczyk, 1991; Chen and Garba, 1979), theproposed methodology focuses on the modeling of the individual components and kinematicjoints that comprise these multibody structures since their dynamic properties play a crucialrole in the dynamic behavior of the complete assembly. Novel and precise models for theprecision stages and bearings have been introduced and incorporated within the multibodyformulation to predict the dynamic response of the positioning systems.

2. DESCRIPTION OF THE POSITIONING SYSTEMS

In this section, two typical positioning systems commonly used at the APS site are described.The optical table is widely used in the positioning and support of synchrotron radiationinstruments. The mirror support system is a brand-new design to support and positionhorizontally deflecting mirrors to provide branching in the insertion device beamlines.

2.1. Optical Thble

A typical optical table assembly is shown in Figure 1. It consists of an optical breadboardattached via three self-aligning ball bearings to three fine-motion vertical lifting stages. These



197

Figure 1. Photograph of a typical APS optical table.

stages rest on horizontal motion guides, which are driven by electric motors. A pseudo- &dquo;cone-flat-v&dquo; type of kinematic mount configuration is used to restrain the optical breadboard. Thistype of configuration provides hand-powered course vertical motion control (178 mm range, 5micron resolution) and motor-powered fine-motion control in five directions, three rotationaland two translational (on the order of tenths of microns and microradians resolution), throughprecision linear and spherical bearings.

2.2. Mirror Support System

The mirror support system consists of three main shafts attached to three precision horizontalguides arranged in a 3-point kinematic mount fashion, cone-flat-v configuration (Shu et al.,1997). The mirror is mounted on a supporting platform. This platform and a parallel lowerplatform are attached to the shafts using self-aligning ball bearings. BMcal translationalmotion and two rotations are added to the design through the use of vertical lifting stagesas in the optical table case. These stages are located below the horizontal guides and drivenby precision worm gear sets powered by electric motors. Figure 2 is a photograph of theuncompleted support assembly taken during the vibration tests. It provides vertical motioncontrol in a 60 mm range with 0.3 micron resolution through a linear screw mechanism.



198

Figure 2. Photograph of the mirror support system.



199

Figure 3. Schematic of the self-aligning ball bearing.

3. COMPONENT STUDIES

In this section, typical components of the optical positioning systems are analyzed bothexperimentally and/or theoretically. Our primary objective is to improve the models forthe calculation of stiffness and damping properties of the complex system joints so that theoverall system response can be predicted more accurately. The components typical to opticalpositioning systems are self-aligning ball bearings, vertical positioning stages, and horizontallinear guides. Main shafts used in the mirror support system design are also analyzed in thissection. The analyses presented will provide a justification for some of the assumptions madein developing the linearized multibody model presented in Section 4.

3.1. Self-Aligning Ball Bearings

Self-aligning ball bearings are standard components of optical positioning systems. As shownin Figure 3, two rows of roller balls are mounted on a fairly standard inner race and an outerrace with a center of curvature coincident with the bearing’s geometric center. Relative to theother system components, the inertial properties of the bearings are negligible. However, thebearings are significant sources of compliance and damping. Bearing stiffness is a nonlinearfunction of externally applied axial and radial force.

Consider the self-aligning double-row ball bearing shown in Figure 4. The relationshipsbetween the mean forces Fun = cm Fym Fzm ]~ transmitted through the bearing and theresulting mean bearing displacements qm = [qxm qym qzm T are derived in detail in Roystonand Basdogan (1998). Highlights of the derivation and relevant results are summarized here.From the bearing displacements, the resultant elastic deformation 6 (If/;) of the jth rollingelement of the ith row located at angle ’If; from the x axis can be determined. Assuming theouter ring is fixed, in Figure 4b, we have



200

Here, Ao and A (y) ) are the unloaded and loaded relative distances between the innerand outer raceway groove curvature centers. Note that Ao = r; , the radius of the locus of theinner raceway groove curvature centers. The axial and radial displacements of the ith row,jth rolling element inner raceway groove curvature center have been denoted by 6 §j and 61 rj ,respectively.

Equation (1), in conjunction with the Hertzian contact stress principle stated as follows,yields the load-deflection relationships for a single rolling element:

Here, Qj~ is the resultant normal load on the rolling element, and Kn is the effective

stiffness constant for the inner race-rolling element-outer race contacts and is a function ofthe bearing geometry and material properties. The exponent n = 3/2 is for a ball bearing(Harris, 1984).

The bearing stiffness matrix is a global representation of the bearing kinematic and elasticcharacteristics as it combines the effects of 2z number of loaded rolling element stiffnessesin parallel. Through vectoral sums Q~ (i = 1, 2, j = x, y, z) in equation (2) for all of theloaded rolling elements, one can relate the resultant bearing mean load vector Fun to thedisplacement vector qm:

where a~ is the loaded contact angle.Now we define a symmetric bearing stiffness matrix Ksb of dimension 3 from equation

(3):

Each stiffness coefficient must be evaluated at the mean point qm. Note that Ksb is

symmetric (i.e., %k = kkj ), and it may be fully populated. This matrix can be written in

terms of its elements as

Using equation (5), the coefficients kps can be directly computed given the mean bearingdisplacement vector qm. However, usually it is the preload vector Fm that is known, not the



201

Figure 4. Spherical bearing: (a) coordinate system, (b) elastic deformation of a rolling element fornonconstant contact angle aj.



202

resulting mean displacement vector. If this is the case, coupled nonlinear equation (3) maybe numerically solved to find qm for a given M. A suitable approach is to use a Newton-Raphson method. _

3.2. krfical Positioning

The fine-motion vertical stages (see Figure 5) were designed by the Experimental FacilitiesDivision of the APS (Shu and Barraza, 1996). These independent lifting stages consist of aninner shell shaped like a cylinder, which is actuated by a precision worm gear set poweredby an electric motor. The inner shell translates along the three sets of linear cylindrical rollerbearings inside of a box-shape outer shell. Theoretical estimates of damping and stiffnessproperties are not available due to the numerous components with complex geometries thatmake up the stage.

At earlier stages of this study, the inner and outer housings of the stages were assumed tobe rigid (Basdogan et al., 1997). Based on this assumption, for the horizontal direction, onlythe compliance of the linear rolling bearings was considered in the multibody formulation.After comparison of theoretical and experimental results, this assumption was found to beinsufficient for the calculation of the modes associated with the horizontal motion of the

optical breadboard. In addition to the linear bearings, compliances of the inner and outerhousings need to be considered. This requires a detailed study of the interface components,which involves a finite element formulation of the bearings and housings. In the followingsubsections, the procedure followed in the calculation of a lumped-mass and compliancemodel for the vertical stage is presented.

3.2. l. Stiffness of the linear roller bearings. To calculate the contact stiffness of the linear rollingbearings, the Hertz theory of elastic contact is used (Flugge, 1962). For a smooth circularcylinder of length lr between two rigid planes and loaded by a force Fr, the displacement ofthe initial contact can be given by the formula

where Re and E, are the equivalent radius and modulus of elasticity of the system,respectively. The radial force Fr on the rolling bearings is calculated based on the resultingforce distribution due to the configuration of the rolling bearings around the inner housingof the actuator. Stiffness coefficient kcyl is the linearized ratio between applied force anddisplacement for a given preload condition.

3.2.2. Stiffness of the lifting mechanism. The stiffness coefficient of the vertical stage in thez-direction is analyzed in two critical sections. The first one will be the point contact of thecoupling between the lifting screw and the top plate of the actuator, and the other one is thelifting screw itself (see Figure 5b).

Point contact stiffness: The deflection of the system due to elastic deformation of thebodies at the contact interface is (Slocum, 1991) the following:



203

Figure 5. Vertical stage: (a) overall schematic, (b) detailed drawing of the lifting mechanism.



204

where Re and Ee are the equivalent radius and modulus elasticity of the system, respectively.As in the previous case, the amount of force (F) exerted on that specific actuator is neededfor the calculation of the stiffness coefficient.

Lifting screw stiffness: Assuming axial deformation along the screw, the stiffnesscoefficient is given by

where ~4~cw is the cross-section area, Escrow is the modulus of elasticity, and Lscrew is the

length of the screw. The point contact stiffness and the lifting screw stiffness will be treatedas springs connected in a series, as shown in Figure 5b.

3.2.3. Global stiffness of the stage. A detailed finite element model (FEM) was constructed thataccounts for the compliance of the inner and outer shells of the vertical stages. Linear rollingbearings are included in the FEM analysis as spring elements having stiffness coefficientsbased on the calculations presented in Section 3.2.1. Solid models for the inner and outer

shells of the vertical stage are created and meshed with four-node, 3-D structural elements.To eliminate the rigid body modes, the base of the outer housing is assumed to be fixed tothe ground. The inner housing and outer housing are interconnected with spring elements torepresent the linear roller bearings. The lifting screw and the coupling are represented witha linear actuator element (see Section 3.2.2 for details), which is also used to connect theinner housing and outer housing in the vertical direction. After the model creation had beencompleted, a static analysis was performed to calculate the deformation of the inner and outerhousing with respect to the fixed ground under the effect of static loads. As shown in Fig-ure 6, the forces are applied in each direction, separately, at the center point of the top plateof the inner housing. The nodal displacements are calculated and recorded at each run. Inthe remainder of this section, the procedure for deriving the equivalent 6 x 6 stiffness matrixfrom the nodal displacements is outlined.

The flexibility influence coefficient a is defined as the displacement at a node due toa unit-applied force (Thomson, 1981). With forces Fx, Fy, and Fz and moments Mx, My,and Mz applied at that node, the principle of superposition can be used to determine thedisplacements in terms of the flexibility influence coefficients. Expressed in matrix form, therelationship between the nodal displacements and nodal forces can be written in terms of theflexibility matrix coefficients as



205

Figure 6. Finite element model of the vertical stage.

or

where a is the flexibility matrix, u is the vector of nodal displacements, and f is the vectorof nodal forces and moments. The general rule for establishing the flexibility elements ofany column using the FEM is to set the force corresponding to that column to unity, with allother forces equal to zero, and calculate the displacements at each degree of freedom of thatspecific node.

A reduced 6 x 6 stiffness matrix can be calculated by taking the inverse of the flexibilitymatrix a. The coefficients of the stiffness matrix can be used to determine generalizedconstants for the springs that connect the inner and outer housing in the linearized multibodyformulation.

3.3. Horizorctal Linear Motion Guides

The linear motion guides consist of rails with a rectangular cross-section and rectangular box-shape carriages that contain passages for recirculating balls. For two-axis motion, typically



206

Figure 7. Schematic of the horizontal linear motion guides.

there are four rails and four carriages (see Figure 7). The pseudo-cone-flat-v kinematicconfiguration of the overall positioning system is achieved by restraining motion along theslides with electric motors in the required directions. The linear motion guides representnegligible inertia but possess significant damping and compliance. No accurate theoreticalmodel for their damping and stiffness properties exists. To predict their stiffness properties,experimental studies have been conducted.

To measure the stiffness of the horizontal guides, they are mounted to an optical tableto perform the tests. A fixture has been designed to apply a static force gradually, and thedisplacement of the surface is measured through the use of an optical encoder attached tothe upper plate of the horizontal motion guide (see Figure 8). The magnitude of the force ismeasured with load cells attached to the upper plate surface. Figure 9 shows the relationshipbetween the applied force and the displacement in each direction. Stiffness coefficients inthe x- and y-directions can be obtained by applying a linear curve fit to the experimentaldata. These values can be integrated into the system equations (see modeling assumptions inSection 4.2) as spring constants (6 x 6 stiffness matrix). The stiffness in the z-directionand the rotational stiffness coefficients cannot easily be obtained experimentally. Thesecoefficients are estimated using the Hertz theory of elastic contact. Their order of magnitudeis significantly higher than the other stiffness coefficients in the system, so their effect isnegligible in the low-frequency band of interest. As can be seen from Figure 9, the stiffnessin the x-direction is higher than that in the y-direction, which could be explained by the motorin the x-direction being fixed to the optical table, whereas in the y-direction, it is mounted tothe upper plate. The tests were repeated several times to verify the consistency of the test.The stiffnesses used in the equations of motion are the averages of the values in Table 1.



207

Table 1. Stiffness coefficients obtained in repeated tests.

3.4. Main Shafts of the Mirror Support System

These shafts are unique to the mirror support system introduced in Section 2.2. They carry thetranslational and rotational motion to the mirror. They also provide the path for the coolingtube to reach the mirror in the vacuum chamber. They play a significant role in the vibratorybehavior of the mirror support system. To estimate their stiffness properties, a FEM has beendeveloped for the hollow shaft. As shown in Figure 10, the shaft is fixed at the end wherethe support post is inserted in the shaft. Static forces are applied at the connection points (1)and (2), where the shaft is attached to the lower and upper platforms, respectively. The nodaldisplacements are calculated at each run to estimate a stiffness matrix for the shaft. F1 andF2 are defined as

A procedure similar to Section 3.2.3 has been followed to estimate the stiffness matrix forthe shaft. The shaft is assumed to be fixed to the vertical stage, and the relative displacementof points (1) and (2) with respect to the stage is predicted. The relationship between the nodaldisplacements and nodal forces can be written as

or

where a is the flexibility matrix, u is the vector of nodal displacements, and f is the vectorof nodal forces. The flexibility matrix is used in the modeling of the joints in the multibodyformulation presented in the following section.



208

Figure 8. Experimental setup for measuring stiffness properties of the horizontal linear motion guides:(a) top view, (b) front view.

_

~t~&dquo; &dquo; It.l’ .

°



209

Figure 9. Displacement of the upper plate of the horizontal guide as a function of applied force.

Figure 10. Finite element model of the main shaft of the mirror support system.



210

4. LINEARIZED MULTIBODY SYSTEM FORMULATION

The dynamic equations of motion of multibody systems are in general nonlinear because ofthe finite rotations of the bodies in the system. In the optical positioning systems examined inthis investigation, the system components experience very small rotations. As a consequence,a linearized set of equations can be obtained. These equations are used to predict the naturalfrequencies and mode shapes of the positioning systems. The results obtained using themultibody model are compared with the results obtained experimentally.

4.1. Dynamic Equations of Motion

A multibody system consisting of nb unconstrained rigid bodies has 6 x nb independentgeneralized coordinates. The vector q of the generalized coordinates of the multibody systemis then defined as

where

The dynamic equations of motion of a rigid body, having the origin of the body coordinatesystem rigidly attached to the center of mass, can be written in a partitioned matrix form as(Shabana, 1994) the following:

where nb is the total number of rigid bodies in the system. The submatrices of the mass matrixassociated, respectively, with the translation and orientation coordinates are m6 and nice.Also, (Q’ v ),6 is the centrifugal force vector. ( Q§ )R and (GZe)~ are the vectors of generalizedforces associated, respectively, with the generalized translation and orientation coordinates.

4.2. Multi-Degree-of-Freedom Optical Positioning Systems

The main steps for the construction of the linearized dynamic equations of motion of theoptical positioning systems will be discussed in this section. These systems are modeledas multibody structures since they consist of interconnected components that undergotranslational and rotational displacements through the use of kinematic joints. The developed



211

methodology is applied to the optical table and mirror support system introduced in Sections2.1 and 2.2. To highlight the modeling details, the multibody models used for the optical tableand mirror support system are introduced here together with the following assumptions (seeFigure 11).

1. Linear system theory is valid (only very low-amplitude motion is expected).2. The breadboard of the optical table is treated as rigid. This was verified experimentally

for the frequency range of interest.3. A lumped-mass and compliance model for the vertical stage has been developed using

the finite element techniques (see Section 3.2.3).4. Bearing elements are treated as compliant and massless (3 x 3 stiffness matrix can be

derived for the axial-only preload condition based on the theory in Section 3.1 ).5. Horizontal linear guides are treated as springs connecting the outer housing to the fixed

ground in the optical table case (see Section 3.3). The horizontal guides are modeleddifferently for the cone, flat, and v joints since different directions are restrained at eachjoint.

6. Horizontal linear guides are treated as springs connecting the inner housing to the mainshafts in the mirror support system case (see Section 3.3). The horizontal guides aremodeled differently for the cone, flat, and v joints since different directions are re-strained at each joint.

7. A lumped-mass and compliance model for the main shafts of the mirror support systemhas been developed using the finite element techniques.

8. Upper and lower platforms of the mirror support system are treated as rigid.9. The support leg of the mirror support system has been considered as a separate rigid

body connected via springs to the experiment floor. Spring coefficients are calculatedbased on the studies by Royston (1998).

The mass matrix of equation (15) is evaluated from the mass and inertia properties ofthe rigid bodies that comprise the system. Stiffness coefficients for the flexible componentsderived in the previous sections are integrated into the system equations to evaluate thegeneralized forces.

4.2.1. Definition of the generalized forces. The forces associated with the orientation

coordinates (Cae)~ and the centrifugal forces ((a;, )~ will be calculated using the multibodyformulation (Shabana, 1994). The vector (Q§ )R in equation (15) represents the effect ofthe external forces applied to a set of points on the system. For the proposed models, theseare the spring forces resulting from the flexibility of the elastic components of the system.These spring forces can be expressed in terms of the generalized coordinates to define thegeneralized forces in the system equations (Basdogan et al., 1996).

4.2.2. Stiffness and mass matrices. All of the generalized forces and moments are evaluatedand combined in the general form of equations of motion that govern the vibration of a multi-degree-of-freedom system for the free undamped case. This equation can be written as



212

Figure 11. Schematic of the theoretical models of the (a) optical table and (b) mirror support system.



213

where the generalized coordinate vector q is defined in equation (14). Here, q consists ofthe independent translational and rotational coordinates of the rigid bodies. K is the globalstiffness matrix and defined as a square matrix having a dimension of 6 x nb (nb : numberof rigid bodies). After the mass and stiffness matrices are evaluated, the natural frequenciesand mode shapes of the system can be calculated using the standard eigenvalue approach(Shabana, 1991).

5. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS

In this section, the theoretical results based on the multibody formulation of the optical tableand mirror support system are compared with experimental measurements. Both assembliesare analyzed experimentally using modal analysis. The frequency response up to 200 Hzis studied using random excitation from an electrodynamic shaker and a fixed excitationapproach. Excitation sources in all directions are used to ensure that all possible vibrationmodes are found.

5.1. Optical Table

The comparison of predicted mode shapes and natural frequencies with experimental studiesis highlighted in Figure 12. Rotational motion of the optical breadboard about the x,y, and z axes is evident. The relevant low-frequency modes found by theory comparewell with experimental results. While the theoretical model is capable of capturing all ofthe significant modes under 80 Hz, it is unable to predict the mode shapes above 80 Hzbecause these are associated with flexural motion of the coarse lifting stage and supporttruss combination, which are assumed fixed in the theoretical system formulation. Since

significant displacement amplitude is expected to be only associated with the lower frequencymodes, this limitation is acceptable.

5.2. Mirror Support System

Similar studies are carried out to predict the dynamic response of the mirror supportsystem described in Section 2.2. Low-frequency theoretically predicted and experimentallymeasured natural frequencies and mode shapes are shown in Figure 13. The first three modesshown in both cases denote rocking conditions in which the entire system is moving as asingle rigid body on the laboratory floor. Theoretical and experimental predictions of the firstrocking mode at 7.6 Hz and the first two deformation modes at 47.1 Hz and 71.6 Hz comparewell. Comparisons for the other two rocking modes in the 10 to 30 Hz range are without goodagreement in natural frequency values. This is believed to be due to an insufficiently accuratemodel for the actual lab floor dynamics. The floor was 8-inch thick reinforced concrete

supported on columns. At the APS beamlines, the mirror support system will be mounted onan 18-inch thick concrete slab on a grade, and it is expected that under these circumstances,system-natural frequencies will be higher. This situation is simulated by using an infinitefloor stiffness value in the multibody model. The resulting natural frequencies (see Table 2)are significantly higher, suggesting that this mirror system design will be adequate.



214

Figure 12. Optical table results for system natural frequencies and mode shapes below 80 Hz: (a)experimental results, (b) theoretical results. :

, ’

,, ,

’ ° °

Figure 13. Mirror support system results for system natural frequencies and mode shapes below 80 Hz:

(a) experimental results, (b) theoretical results.



215

Table 2. Predicted natural frequencies with rigid floor. ---

6. DISCUSSION AND CONCLUSION

A methodology has been developed to predict the vibratory response of typical high-precisionoptical positioning systems. The methodology has been applied to specific example cases-an optical table and a mirror support system-commonly used at the APS site. Some ofthe positioning system components were treated as rigid while others required a flexibleformulation. Comparison of theory to supporting experimental studies illustrated the crucialimportance of having accurate models for the various joint and linkage components. Detailedexperimental and theoretical models have been developed for the kinematic joint componentssuch as vertical and horizontal stages, as well as linear and rotational bearings. These modelshave been incorporated into the multibody formulation and then generalized for an unlimitedrange of positioning configurations, given the orientation of the rigid bodies. After the

dynamic equations had been formulated, the resonance frequencies and associated modeshapes were presented for sample configurations.

Experimental and theoretical studies are in progress to quantify the levels and frequenciesof existing vibration sources, such as seismic and facility-bome vibrations and flow-induced vibration in the cooling systems of the optical devices. In addition, the dampingcharacteristics of the whole structure and some of the individual components remain to be

quantified for complete system modeling. Then, the developed model will be used to simulatethe vibratory response of the positioning systems to typical vibratory excitations seen in day-to-day use.

The proposed methodology can be applied to similar systems at the APS and otherfacilities. It will assist designers by enabling them to investigate the system componentsindividually and as a part of the assembled system. It is expected that it will facilitate futuredesign improvements for the next-generation beamlines at the APS and similar facilities.

Acknowledgment. This work was supported by the U.S. Department of Energy BES-Materials Sciences, undercontract no. W-31-109-ENG-38.

REFERENCES

Anees, A., 1995, "Interactive simulation-based optical system development," SPIE Proceedings CR58, 220-248.Baklakov, B. A., Lebedev, P. K., Parkhomchuk, V V, Seryi, A. A., Sleptsov, A. I., and Shil’tsev, V D., 1993, "Study of

seismic vibrations for the VLEPP linear collider: Technical physics," Technical Physics 38, 894-898.



216

Basdogan, I., Royston, T J., Barraza, J., Shu, D., and Kuzay, T, 1997, "A theoretical and experimental study of the vibra-tory behavior of a high-precision optical positioning table," in Proceedings of the ASME Design EngineeringTechnical Conference, Sacramento, CA.

Basdogan, I., Royston, T. J., Shabana, A. A., Shu, D., and Kuzay, T, 1996, "Vibratory response of a mirror sup-port/positioning system for the Advanced Photon Source Project at Argonne National Laboratory," SPIE Pro-ceedings 2865, 1-14.

Brochet, A. and Bernard, J., 1991, "Site selection and design of a low-vibration building for submicron technologyapplications," SPIE Proceedings 1619, 148-160.

Chen, J. C. and Garba, J., 1979, "Analytical model improvement using modal test results," AIAA Journal 18(6), 684-690.Doyle, B. K., 1993, "Design of optical structures for maximum fundamental frequency," SPIE Proceedings 1998, 50-59.Doyle, B. K., Cerrati, V J., Forman, S. E., and Sultana, J. A., 1995, "Optimal structural design of the airborne infrared

imager," SPIE Proceedings 2542, 11-22.

Dyck, H. M., Berg, E., and Harris, D., 1986, "A study of microseismic and man-made vibrations on Mauna Kea," SPIEProceedings 628, 162-169.

Flugge, W, 1962, Handbook of Engineering Mechanics, McGraw-Hill, New York.

Harris, T A., 1984, Rolling Bearing Analysis, John Wiley, New York.

Jendrzejczyk, J. A., 1991, "Vibration considerations in the design of the Advanced Photon Source at Argonne NationalLaboratory," SPIE Proceedings, 1619, 114-126.

Jennings, P. A. and Willson, J. W., 1991, "Improved vibration isolation system for a scanning electron microscope,"SPIE Proceedings 1619, 148-160.

Laughlin, M. J., 1995, "Mirror jitter: An overview of theoretical and experimental work," Optical Engineering 34(2),321-329.

Redding, D. C. and Breckenridge, W G, 1991, "Optical modeling for dynamics and control analysis," Journal of Guid-ance, Control, and Dynamics 14(5), 1021-1032.

Royston, T J., 1998, "Vibration characteristics of an APS lab facility in building 401," Argonne National Laboratory,Light Source Notes, http://www.aps.anl.gov/techpubn/Isnotes/Is263/Is263.html.

Royston, T J. and Basdogan, I., 1998, "Vibration transmission through self-aligning (spherical) rolling element bearings:Theory and experiment," Journal of Sound and Vibration 215(5), 997-1014.

Shabana, A. A., 1991, Theory of Vibration, Vol. 2, Springer-Verlag, New York.Shabana, A. A., 1994, Computational Dynamics, Springer-Verlag, New York.Shu, D. and Barraza, J., 1996, "Low profile, high load vertical rolling positioning stage," U.S. patent #5, 5426, 903,

June 18.

Shu, D., Benson, C., Chang, J., Barraza, J., Kuzay, T M., Alp, E. E., Sturhan, W, Lai, B., McNulty, I., Randall, K.,Srajer, G, Xu, Z., and Yun, W, 1997, "Mirror mounts designed for the Advanced Photon Source SRI-CAT," inProceedings of SRI’97, Cornell University, New York.

Slocum, A. H., 1991, Precision Machine Design, Prentice Hall, Englewood Cliffs, NJ.

Thomson, W T, 1981, Theory of Vibration With Applications, Prentice Hall, Englewood Cliffs, NJ.



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